On the involution Jimm (Q2055096)

Summary: This chapter is a survey of the involution of the real line induced by Dyer's outer automorphism of the group \(\mathrm{PGL}(2,\mathbb Z)\). This `modular' involution is discontinuous at the rationals but satisfies a surprising collection of functional equations. It preserves the set of real quadratic irrationals mapping them in a non-obvious way to each other. It commutes with the Galois action on real quadratic irrationals. It restricts to a non-trivial involution of the set of elements of norm \(+1\) in real quadratic number fields. More generally, it preserves set-wise the orbits of the modular group, thereby inducing an involution of the moduli space of real rank-two lattices. We give a description of this involution as the boundary action of a certain automorphism of the infinite trivalent tree. It is conjectured that algebraic numbers of degree at least three are mapped to transcendental numbers under this involution. For the entire collection see [Zbl 1466.57001].